The document describes how complex numbers can be represented geometrically using an Argand diagram. The Argand diagram plots the real components of complex numbers along the x-axis and imaginary components along the y-axis, allowing each complex number to be associated with a unique point on the plane. The document then introduces the modulus-argument form for representing complex numbers, where the modulus is the distance from the origin to the point and the argument is the angle relative to the positive real axis.
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
This document discusses various numerical integration techniques including Newton-Cotes formulas, the trapezoidal rule, Simpson's rules, integration with unequal segments, open integration formulas, integration of equations, and Romberg integration. The key Newton-Cotes formulas covered are the trapezoidal rule, Simpson's 1/3 rule, and Simpson's 3/8 rule. The document provides examples of applying these formulas to numerically evaluate definite integrals and calculates the associated errors. It also discusses using Richardson extrapolation, known as Romberg integration, to iteratively improve the accuracy of numerical integration compared to the standard Newton-Cotes formulas.
This document provides an analysis of the time response of control systems. It defines time response as the output of a system over time in response to an input that varies over time. The time response analysis is divided into transient response, which decays over time, and steady state response. Different types of input signals are described, including step, ramp, and sinusoidal inputs. Methods for analyzing the first and second order systems are presented, including determining the transient and steady state response. Static error coefficients like position, velocity and acceleration constants are defined for different system types and inputs. Examples are provided to illustrate the analysis of first and second order systems.
This document discusses phase lead and lag compensators for digital control systems. It covers:
1. Designing a discrete-time phase lead/lag compensator by mapping the z-plane to the w-plane using bilinear transformation.
2. Defining phase lead and lag compensators based on the positions of poles and zeros in the w-domain transfer function.
3. A design approach using frequency response methods to meet a phase margin specification by determining the parameters of a first-order digital phase lead or lag compensator.
4. Examples of designing phase lead and lag compensators for different plant transfer functions to meet specifications on phase margin and steady state error.
This document discusses impulse response in signals and systems. It defines an impulse signal as having a value of zero except at t=0, where it has an infinitely high value. The impulse response describes the output of a system when given an impulse as input. It provides an example of finding the poles and zeros of a simple system transfer function. The document also derives the impulse response and step response of a first order system and explains the relationship between the two responses. Impulse response has applications in areas like loudspeakers, audio processing, and control systems.
1) Rectangular waveguides can transmit electromagnetic waves above a certain cutoff frequency, acting as a high-pass filter. They support transverse electric (TE) and transverse magnetic (TM) modes of propagation.
2) For TM modes, the electric field is transverse to the direction of propagation, while the magnetic field has a longitudinal component. The modes are denoted TMmn, with m and n indicating the number of half-wavelength variations across the width and height.
3) For TE modes, the magnetic field is entirely transverse, while the electric field has a longitudinal component. These modes are denoted TEmn, with m and n having the same meaning as in the TM case.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
This document discusses various numerical integration techniques including Newton-Cotes formulas, the trapezoidal rule, Simpson's rules, integration with unequal segments, open integration formulas, integration of equations, and Romberg integration. The key Newton-Cotes formulas covered are the trapezoidal rule, Simpson's 1/3 rule, and Simpson's 3/8 rule. The document provides examples of applying these formulas to numerically evaluate definite integrals and calculates the associated errors. It also discusses using Richardson extrapolation, known as Romberg integration, to iteratively improve the accuracy of numerical integration compared to the standard Newton-Cotes formulas.
This document provides an analysis of the time response of control systems. It defines time response as the output of a system over time in response to an input that varies over time. The time response analysis is divided into transient response, which decays over time, and steady state response. Different types of input signals are described, including step, ramp, and sinusoidal inputs. Methods for analyzing the first and second order systems are presented, including determining the transient and steady state response. Static error coefficients like position, velocity and acceleration constants are defined for different system types and inputs. Examples are provided to illustrate the analysis of first and second order systems.
This document discusses phase lead and lag compensators for digital control systems. It covers:
1. Designing a discrete-time phase lead/lag compensator by mapping the z-plane to the w-plane using bilinear transformation.
2. Defining phase lead and lag compensators based on the positions of poles and zeros in the w-domain transfer function.
3. A design approach using frequency response methods to meet a phase margin specification by determining the parameters of a first-order digital phase lead or lag compensator.
4. Examples of designing phase lead and lag compensators for different plant transfer functions to meet specifications on phase margin and steady state error.
This document discusses impulse response in signals and systems. It defines an impulse signal as having a value of zero except at t=0, where it has an infinitely high value. The impulse response describes the output of a system when given an impulse as input. It provides an example of finding the poles and zeros of a simple system transfer function. The document also derives the impulse response and step response of a first order system and explains the relationship between the two responses. Impulse response has applications in areas like loudspeakers, audio processing, and control systems.
1) Rectangular waveguides can transmit electromagnetic waves above a certain cutoff frequency, acting as a high-pass filter. They support transverse electric (TE) and transverse magnetic (TM) modes of propagation.
2) For TM modes, the electric field is transverse to the direction of propagation, while the magnetic field has a longitudinal component. The modes are denoted TMmn, with m and n indicating the number of half-wavelength variations across the width and height.
3) For TE modes, the magnetic field is entirely transverse, while the electric field has a longitudinal component. These modes are denoted TEmn, with m and n having the same meaning as in the TM case.
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
The document discusses the design of discrete-time IIR filters from continuous-time filter specifications. It covers common IIR filter design techniques including the impulse invariance method, matched z-transform method, and bilinear transformation method. An example applies the bilinear transformation to design a first-order low-pass digital filter from a continuous analog prototype. Filter design procedures and steps are provided.
This document discusses sampling and related concepts in signal processing. It begins by introducing the need to convert analog signals to discrete-time signals for digital processing. It then covers the sampling theorem, which states that a band-limited signal can be reconstructed if sampled at twice the maximum frequency. The document describes three main sampling methods: ideal (impulse), natural (pulse), and flat-top sampling. It also discusses aliasing, which occurs when a signal is under-sampled. The key aspects of sampling covered are the sampling rate, reconstruction of sampled signals, and anti-aliasing filters.
Fundamentals of Digital Signal Processing - Question BankMathankumar S
This document contains questions related to the fundamentals of digital signal processing. It covers topics such as digital signal processing concepts, z-transforms, discrete Fourier transforms, digital filter design including IIR and FIR filters, and multi-rate signal processing. The questions range from conceptual definitions and properties to practical problems involving calculations and filter design. The document contains questions to test knowledge of key DSP topics and techniques.
The Fourier transform decomposes a signal into its constituent frequencies, representing it in the frequency domain rather than the spatial domain, which can make certain operations and analyses easier to perform; it has both magnitude and phase components that provide information about the frequency content and relative phases of the signal. The discrete Fourier transform (DFT) is a sampled version of the continuous Fourier transform that is useful for digital signal and image processing applications.
State space analysis shortcut rules, control systems, Prajakta Pardeshi
This document discusses different types of state space analysis including physical variable form, phase variable form using canonical forms I and II, parallel realization, converting between state models and transfer functions, state transition matrices, and observability and controllability. It provides examples of obtaining state space models from electrical circuits using different approaches like writing standard state equations, using canonical forms, and parallel realization from transfer functions. It also outlines how to check for observability and controllability of state space models.
The document provides examples of block diagram reduction techniques. It begins with 12 examples showing the step-by-step process of reducing block diagrams to obtain transfer functions. It then provides block diagram examples of an armature controlled DC motor and a liquid level system. The examples illustrate techniques for moving pickoff points, eliminating feedback loops, and applying rules to reduce block diagrams to single transfer functions.
This document contains solved problems related to digital communication systems. It begins by defining key elements of digital communication systems such as source coding, channel encoders/decoders, and digital modulators/demodulators. It then solves problems involving Fourier analysis of signals and generalized Fourier series. The problems cover topics like measuring performance of digital systems, classifying signals as energy or power, sketching signals, and approximating signals using generalized Fourier series.
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
Bilinear z-transformation is the most common method for converting the transfer function H(s) of the analog filter to the transfer function H(z) of the digital filter and vice versa. In this work, introducing the relationship between the digital coefficients and the analog coefficients in the matrix equation definitely involves the Pascal’s triangle.
Power electronics devices and their characteristicsKartickJana3
This document discusses power electronic devices and their characteristics. It describes several types of power devices including bipolar junction transistors (BJT), field effect transistors (FET), thyristors, Darlington transistors, and insulated gate bipolar transistors (IGBT). It covers the key characteristics, operating principles, and ratings of these devices. It also discusses how snubber circuits using inductors, resistors, and capacitors can be designed to protect power devices from high rates of change of current (di/dt) and voltage (dv/dt) during switching.
The document discusses PID tuning methods, including Ziegler-Nichols tuning rules. It provides two Ziegler-Nichols methods for determining PID parameters based on experimental step response data or critical gain/period. The first method uses delay time and time constant from a step response. The second method varies proportional gain until sustained oscillations occur, then uses critical gain and period. An example applies the second method to tune a PID controller for a plant, showing it results in excessive overshoot requiring fine tuning to reduce overshoot to 25% or less.
This document provides an introduction to feedback control systems for agricultural engineering. It discusses key concepts in control systems including open vs closed loop control, continuous vs sequential control, and control system components like sensors, controllers, and actuators. Different types of control systems are described for applications like machine control, process control, analog systems, and digital systems. Terminology used in control systems like controlled variables, manipulates variables, and disturbances are also defined. The document outlines topics that will be covered, including modeling, analysis, design, and mathematical methods like Laplace transforms, transfer functions, and linearization.
classification of second order partial differential equationjigar methaniya
This active learning assignment discusses the classification of second order partial differential equations. The general form of a non-homogeneous second order PDE is presented. A PDE is classified as elliptic if B^2 - 4AC < 0, parabolic if B^2 - 4AC = 0, and hyperbolic if B^2 - 4AC > 0. Three examples are worked through to demonstrate classifying PDEs as elliptic, parabolic, and hyperbolic by comparing them to the general form.
This document provides an overview of transfer functions and stability analysis of linear time-invariant (LTI) systems. It discusses how the Laplace transform can be used to represent signals as algebraic functions and calculate transfer functions as the ratio of the Laplace transforms of the output and input. Poles and zeros are introduced as important factors for stability. A system is stable if all its poles reside in the left half of the s-plane and unstable if any pole resides in the right half-plane. Examples are provided to demonstrate calculating transfer functions from differential equations and analyzing stability based on pole locations.
This document contains 20 multiple choice questions from a GATE EE exam. It covers topics in signals and systems, circuits, electromagnetic theory, machines, and instrumentation. For each question, the full question and multiple choice options are provided, along with the solution and explanation for the correct answer.
Fourier analysis of signals and systemsBabul Islam
This document discusses Fourier analysis of signals and linear time-invariant (LTI) systems. It defines LTI systems and explains that they are mathematically easy to analyze due to properties like superposition. Fourier analysis is used to represent signals in the frequency domain using techniques like the Fourier series for periodic signals and the Fourier transform for aperiodic signals. The frequency response of an LTI system is its output when the input is an impulse, and the output of any LTI system is the convolution of the input signal and impulse response.
Z-transforms can be used to analyze systems described by difference equations. The Z-transform relates the terms of a discrete-time sequence to a complex function of a complex variable z. Some key applications of Z-transforms include analyzing Fibonacci series, Newton's interpolation formula, and compound interest problems. Important results for Z-transforms include theorems regarding shifting, constants, initial values, final values, and convolution. Z-transforms are also connected to other transforms like the Fourier transform and Laplace transform through relationships like the discrete-time Fourier transform and bilinear transform.
SCRs are mainly used in devices where the control of high power, possibly coupled with high voltage, is demanded. Their operation makes them suitable for use in medium- to high-voltage AC power control applications, such as lamp dimming, power regulators and motor control.
1. The document discusses time domain analysis of second order systems. It defines key terms like damping ratio, natural frequency, and describes the four categories of responses based on damping ratio: overdamped, underdamped, undamped, and critically damped.
2. An example shows how to determine the natural frequency and damping ratio from a given transfer function. The poles of a second order system depend on these parameters.
3. The time domain specification of a second order system's step response is explained, including definitions of delay time, rise time, peak time, settling time, and overshoot.
This document discusses sampling and related concepts in signal processing. It begins by introducing the need to convert analog signals to discrete-time signals for digital processing. It then covers the sampling theorem, which states that a band-limited signal can be reconstructed if sampled at twice the maximum frequency. The document describes three main sampling methods: ideal (impulse), natural (pulse), and flat-top sampling. It also discusses aliasing, which occurs when a signal is under-sampled. The key aspects of sampling covered are the sampling rate, reconstruction of sampled signals, and anti-aliasing filters.
Fundamentals of Digital Signal Processing - Question BankMathankumar S
This document contains questions related to the fundamentals of digital signal processing. It covers topics such as digital signal processing concepts, z-transforms, discrete Fourier transforms, digital filter design including IIR and FIR filters, and multi-rate signal processing. The questions range from conceptual definitions and properties to practical problems involving calculations and filter design. The document contains questions to test knowledge of key DSP topics and techniques.
The Fourier transform decomposes a signal into its constituent frequencies, representing it in the frequency domain rather than the spatial domain, which can make certain operations and analyses easier to perform; it has both magnitude and phase components that provide information about the frequency content and relative phases of the signal. The discrete Fourier transform (DFT) is a sampled version of the continuous Fourier transform that is useful for digital signal and image processing applications.
State space analysis shortcut rules, control systems, Prajakta Pardeshi
This document discusses different types of state space analysis including physical variable form, phase variable form using canonical forms I and II, parallel realization, converting between state models and transfer functions, state transition matrices, and observability and controllability. It provides examples of obtaining state space models from electrical circuits using different approaches like writing standard state equations, using canonical forms, and parallel realization from transfer functions. It also outlines how to check for observability and controllability of state space models.
The document provides examples of block diagram reduction techniques. It begins with 12 examples showing the step-by-step process of reducing block diagrams to obtain transfer functions. It then provides block diagram examples of an armature controlled DC motor and a liquid level system. The examples illustrate techniques for moving pickoff points, eliminating feedback loops, and applying rules to reduce block diagrams to single transfer functions.
This document contains solved problems related to digital communication systems. It begins by defining key elements of digital communication systems such as source coding, channel encoders/decoders, and digital modulators/demodulators. It then solves problems involving Fourier analysis of signals and generalized Fourier series. The problems cover topics like measuring performance of digital systems, classifying signals as energy or power, sketching signals, and approximating signals using generalized Fourier series.
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
Bilinear z-transformation is the most common method for converting the transfer function H(s) of the analog filter to the transfer function H(z) of the digital filter and vice versa. In this work, introducing the relationship between the digital coefficients and the analog coefficients in the matrix equation definitely involves the Pascal’s triangle.
Power electronics devices and their characteristicsKartickJana3
This document discusses power electronic devices and their characteristics. It describes several types of power devices including bipolar junction transistors (BJT), field effect transistors (FET), thyristors, Darlington transistors, and insulated gate bipolar transistors (IGBT). It covers the key characteristics, operating principles, and ratings of these devices. It also discusses how snubber circuits using inductors, resistors, and capacitors can be designed to protect power devices from high rates of change of current (di/dt) and voltage (dv/dt) during switching.
The document discusses PID tuning methods, including Ziegler-Nichols tuning rules. It provides two Ziegler-Nichols methods for determining PID parameters based on experimental step response data or critical gain/period. The first method uses delay time and time constant from a step response. The second method varies proportional gain until sustained oscillations occur, then uses critical gain and period. An example applies the second method to tune a PID controller for a plant, showing it results in excessive overshoot requiring fine tuning to reduce overshoot to 25% or less.
This document provides an introduction to feedback control systems for agricultural engineering. It discusses key concepts in control systems including open vs closed loop control, continuous vs sequential control, and control system components like sensors, controllers, and actuators. Different types of control systems are described for applications like machine control, process control, analog systems, and digital systems. Terminology used in control systems like controlled variables, manipulates variables, and disturbances are also defined. The document outlines topics that will be covered, including modeling, analysis, design, and mathematical methods like Laplace transforms, transfer functions, and linearization.
classification of second order partial differential equationjigar methaniya
This active learning assignment discusses the classification of second order partial differential equations. The general form of a non-homogeneous second order PDE is presented. A PDE is classified as elliptic if B^2 - 4AC < 0, parabolic if B^2 - 4AC = 0, and hyperbolic if B^2 - 4AC > 0. Three examples are worked through to demonstrate classifying PDEs as elliptic, parabolic, and hyperbolic by comparing them to the general form.
This document provides an overview of transfer functions and stability analysis of linear time-invariant (LTI) systems. It discusses how the Laplace transform can be used to represent signals as algebraic functions and calculate transfer functions as the ratio of the Laplace transforms of the output and input. Poles and zeros are introduced as important factors for stability. A system is stable if all its poles reside in the left half of the s-plane and unstable if any pole resides in the right half-plane. Examples are provided to demonstrate calculating transfer functions from differential equations and analyzing stability based on pole locations.
This document contains 20 multiple choice questions from a GATE EE exam. It covers topics in signals and systems, circuits, electromagnetic theory, machines, and instrumentation. For each question, the full question and multiple choice options are provided, along with the solution and explanation for the correct answer.
Fourier analysis of signals and systemsBabul Islam
This document discusses Fourier analysis of signals and linear time-invariant (LTI) systems. It defines LTI systems and explains that they are mathematically easy to analyze due to properties like superposition. Fourier analysis is used to represent signals in the frequency domain using techniques like the Fourier series for periodic signals and the Fourier transform for aperiodic signals. The frequency response of an LTI system is its output when the input is an impulse, and the output of any LTI system is the convolution of the input signal and impulse response.
Z-transforms can be used to analyze systems described by difference equations. The Z-transform relates the terms of a discrete-time sequence to a complex function of a complex variable z. Some key applications of Z-transforms include analyzing Fibonacci series, Newton's interpolation formula, and compound interest problems. Important results for Z-transforms include theorems regarding shifting, constants, initial values, final values, and convolution. Z-transforms are also connected to other transforms like the Fourier transform and Laplace transform through relationships like the discrete-time Fourier transform and bilinear transform.
SCRs are mainly used in devices where the control of high power, possibly coupled with high voltage, is demanded. Their operation makes them suitable for use in medium- to high-voltage AC power control applications, such as lamp dimming, power regulators and motor control.
1. The document discusses time domain analysis of second order systems. It defines key terms like damping ratio, natural frequency, and describes the four categories of responses based on damping ratio: overdamped, underdamped, undamped, and critically damped.
2. An example shows how to determine the natural frequency and damping ratio from a given transfer function. The poles of a second order system depend on these parameters.
3. The time domain specification of a second order system's step response is explained, including definitions of delay time, rise time, peak time, settling time, and overshoot.
This document discusses triggers that can lead to unwanted sexual behavior and how to manage them. It covers sexual triggers, non-sexual environmental triggers like certain rooms or people, and non-sexual emotional triggers like a lack of self-differentiation. It emphasizes that sobriety must come before all else and provides strategies like avoiding triggers, being aware of their existence, and listing specific triggers and potential remedy actions to address them. The goal is to help people make real changes through sobriety rather than just white-knuckle changes by managing triggers that could undermine recovery efforts.
XEN-TAN (pronounced ZEN-TAN) is the revolutionary new sunless tanning range that’s taking the UK and Ireland by storm.
A complete breakthrough in fake tanning, Our XEN-TAN experts have worked very hard to provide you with the complete sunless tanning solution.
XEN-TAN has overcome all the dreaded drawbacks so frequently associated with most sunless tanning products on the market:
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What you should do in the following situationsF El Mohdar
If a person finds dirt on their hands while making wudoo', it does not break their wudoo' as long as they are working to remove it as part of purification. If they have a wound that cannot be washed or covered, they should do wudoo' and tayammum for the wounded area. Finding traces of janabah on clothes after praying does not invalidate the prayers as long as the clothes were pure during the prayers.
The document defines angular velocity as the rate of change of the angle swept out by a point moving along a circular path with respect to time. It shows that the linear or tangential velocity of the point is equal to the product of its angular velocity and radius. The period of motion is defined as the time taken for one complete revolution, which is calculated by dividing 2π by the angular velocity. An example calculates the angular velocity and tangential velocity of a satellite in circular orbit.
X2 t03 06 chord of contact & properties (2012)Nigel Simmons
The document discusses properties of ellipses and hyperbolas. It shows that:
1) The chord of contact from a point on the directrix of an ellipse is a focal chord, as it passes through one of the foci.
2) For an ellipse, the part of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.
3) For a rectangular hyperbola, if an external point T is chosen, two tangents can be drawn from T and their points of contact P and Q must both lie on the chord of contact, which is the line PQ. The coordinates of T are also derived in terms of P and Q.
Xabec es una obra corporativa del Opus Dei que promueve la formación y la inserción laboral de técnicos para desempeñar profesiones que requieren una especial cualificación. Está situado en Valencia, y sus orígenes se remontan al centenario del nacimiento de San Josemaría, en 2002
Más información en http://www.xabec.es/
This manual provides important safety information for operating a firearm. It instructs the user to read the manual before handling the firearm and keep the manual with the firearm. It emphasizes following safety rules when using, storing, and transferring firearms and being a responsible firearm owner. The manual then details contents such as safety devices, loading, firing, unloading, cleaning and maintenance instructions.
The document discusses various topics around relationships, marriage, and sexuality. It explores the definition of love from the Bible, recommends books on marriage, and discusses the importance of patience in relationships. It also addresses being purposefully single in order to learn how to be a wife and finding favor through pregnancy within purpose. The next section is listed as being about the Holy Spirit and the feminine aspect of God as the Helper.
La persona estaba preocupada por el bienestar de otra persona y fue a ver qué estaba haciendo, pero se dio cuenta de que no estaba haciendo nada productivo como de costumbre.
Xabi muestra habilidades avanzadas para su edad en comprensión lectora. Reconoce signos gráficos, atribuye significados a palabras, organiza proposiciones, establece relaciones entre ellas, extrae el significado global, asigna categorías y recupera conocimientos previos de manera coordinada y estratégica para construir el significado del texto. Lee con fluidez y velocidad adecuada, realizando pocos errores.
XBRL and the MACPA - Summit PresentationThomas Hood
Slides of a presentation given by myself and Skip Falatko of the MACPA. This presentation covers what XBRL is in broad terms as well as an in-depth look at how the MACPA is using XBRL to improve internal efficiencies as well as extend the U.S GAAP FR Taxonomy.
It is easy to plot a real number in a number line. For example, number 3 is plotted here, and number -2 is plotted on the left side of zero.
we also can plot a two-dimensional number like 2x+3y in an XY plane which is called cartesian coordinate system. By going 3units in the x-axis and 2 units in the y-axis we get the point (2,3) now look at the complex number 5+4i, this number has a real part 5 and an imaginary part 4. now what if we want this point to plot?
It was Jean-Robert Argand who solved the problem and invented an argan plane, which is also called a complex plane. He assumed the x-axis as the real axis and the y-axis as the imaginary axis. so the graphical representation of point 5+4i can be plotted.
so I guess u all are now clear abt what is argand diagram.
yes, graphical representation of a complex number is called argand diagram where the x-axis is the real axis and the y-axis is the imaginary axis.
let z= x+iy where x is the real part and iy is the imaginary part. Point A is the graphical representation of Z. we will be able to understand it more clearly by these examples. A is 3+4i n its representation is shown in the diagram! also, any real number can be shown by argand diagram
real number means the coefficient of i is 0.
the polar form of a complex number can also be shown by the argand diagram. before talking about the polar number, we have to know about two terms. 1 is modulus. The modulus of a complex number is just the magnitude of the vector z since |z| is a length it does not include i. The argument of a complex number written as arg(z), is the angle the vector makes with the positive real axis.
uses: graphical addition and subtraction,
multiplication, representation.
1) The document discusses representing complex numbers geometrically using the Argand diagram. Complex numbers a + ib can be represented as a point (a,b) on the Argand plane, with the real part a on the x-axis and imaginary part b on the y-axis.
2) Examples are given of representing different complex numbers as points on the Argand plane, such as 2 + 3i as point (2,3). It is shown that a + bi is not the same as -a - bi, a - bi, or -z.
3) The modulus (absolute value) of a complex number a + ib is defined as the distance from the point (a,b) representing
The document describes how complex numbers can be represented geometrically on an Argand diagram, which uses a horizontal x-axis for the real part and a vertical y-axis for the imaginary part. It provides examples of various complex numbers plotted as points on the diagram and notes that conjugates are reflections across the x-axis, while opposites involve a 180° rotation. The document then discusses how loci of complex functions can be represented as horizontal and vertical lines, circles, or other shapes on the Argand diagram. It provides two examples, showing that the locus of (i) (z + 4 + i)(z + 4 - i) = 49 is a circle with center (-4, -1) and radius 7 units
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -1.
Aieee 2003 maths solved paper by fiitjeeMr_KevinShah
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -i.
This document contains 20 multiple choice questions about complex numbers. The questions cover topics such as solving complex equations, finding roots of complex polynomials, geometric representations of complex number sets, and manipulating complex expressions. Correct answers are provided for each question.
The document discusses matrices and their properties. It defines minors as the determinants of submatrices formed by deleting a row and column from the original matrix. Cofactors are defined as signed minors, with the sign determined by the parity of the sum of the row and column indices. The adjoint and inverse of matrices are discussed, with the inverse existing only for nonsingular matrices where the determinant is non-zero. Properties of the inverse include that the inverse of the inverse is the original matrix and that the inverse of a product is the product of the inverses in reverse order. Examples are provided to demonstrate calculating the adjoint, inverse and checking the inverse property.
This document contains numerous formulas and theorems from topics including sequences and series, systems of linear equations, quadratic equations, geometry, trigonometry, and calculus. Key formulas include the quadratic formula, distance formulas, slope formula, midpoint formula, Pythagorean theorem, sine and cosine laws, and compound interest formula. Theorems cover properties of angles, circles, and trigonometric identities.
1. The document contains multiple choice questions from chapters 1-3 on real numbers, logarithms, and algebraic expressions & formulas.
2. There are 21 questions total across the three topics testing concepts like properties of real numbers, logarithm rules, and simplifying algebraic expressions.
3. Answer choices are provided for each question to select the correct response.
This document contains a mathematics exam with questions about complex numbers. It has four sections - straight objective type questions with single correct answers, multiple correct answer type questions, matrix matching questions, and a linked comprehension passage with follow up questions. The document tests knowledge of topics like complex number operations, properties of complex functions, geometry of complex numbers on the Argand plane, and relationships between complex number representations of geometric shapes.
This document contains sample questions from a mathematics exam blueprint and marking scheme for Class 12. It includes:
- A blueprint showing the distribution of questions across different units of the syllabus for very short answer (1 mark), short answer (4 marks) and long answer (6 marks) questions.
- Sample questions from sections A to D with varying marks. The questions cover topics like relations and functions, matrices, calculus, vectors, probability and linear programming.
- A marking scheme providing solutions to the sample questions with marks allocated for each step.
This document contains sample questions from a mathematics exam blueprint and marking scheme for Class 12. It includes:
- The exam blueprint which divides the syllabus into 6 units and specifies the number and type of questions to be asked from each.
- Sample questions from Sections A, B, C and D ranging from 1 to 6 marks. The questions cover topics like relations and functions, matrices, calculus, vectors, probability etc.
- The marking scheme which provides the steps and working for solving each question and awards marks accordingly.
1. A complex number can be represented as a sum of a real number and an imaginary number in the form a + bi, where a is the real part and b is the imaginary part.
2. Complex numbers can represent vectors and are useful for representing quantities that involve both magnitude and direction, such as forces.
3. Operations like addition, subtraction, multiplication, and division can be performed on complex numbers by separately applying the operations to the real and imaginary parts.
The document provides an overview of the theory of complex numbers. It defines key concepts such as the Argand diagram, polar and rectangular forms, and operations involving complex numbers like addition, subtraction, multiplication, and division. It also discusses applications of complex numbers to electrical circuits and provides sample exercises for working with complex numbers in different forms and operations.
The inverse of a 2x2 matrix is given by:
A-1 = (1/det(A)) *
[d -b]
[-c a]
Where det(A) is the determinant of A, which is ad - bc. This formula works for any invertible 2x2 matrix.
The document contains 20 multiple choice questions about complex numbers. It tests concepts such as geometric representations of complex numbers and sets, properties of complex functions, solutions to complex equations, and calculations involving complex numbers. The questions range from identifying geometric shapes formed by complex roots to evaluating expressions and solving inequalities involving complex variables.
The document is a math worksheet from Xinmin Secondary School for students in class 3E on graphing exponential and logarithmic functions. It contains 6 questions asking students to sketch various exponential and logarithmic graphs and identify intercepts, asymptotes and solutions to related equations by finding the intersection points of the graphs. Students are reminded to clearly label their graphs and ensure they are large enough.
(1) The document discusses various topics in geometry including lines, circles, triangles, and coordinate geometry. Key concepts discussed include the centroid, orthocentre, and circumcentre of a triangle as well as equations of lines and circles.
(2) Formulas are provided for distances between points and lines, parallel lines, perpendiculars from points to lines, and images of points in lines. Theorems regarding secants and intercepts made by circles on lines are also summarized.
(3) Standard notations used for circles are defined, such as representing the value of the equation of a circle at a point (x1, y1) as S1. Special cases of circles like those touching or passing through
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
What is an RPA CoE? Session 1 – CoE VisionDianaGray10
In the first session, we will review the organization's vision and how this has an impact on the COE Structure.
Topics covered:
• The role of a steering committee
• How do the organization’s priorities determine CoE Structure?
Speaker:
Chris Bolin, Senior Intelligent Automation Architect Anika Systems
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...Jason Yip
The typical problem in product engineering is not bad strategy, so much as “no strategy”. This leads to confusion, lack of motivation, and incoherent action. The next time you look for a strategy and find an empty space, instead of waiting for it to be filled, I will show you how to fill it in yourself. If you’re wrong, it forces a correction. If you’re right, it helps create focus. I’ll share how I’ve approached this in the past, both what works and lessons for what didn’t work so well.
LF Energy Webinar: Carbon Data Specifications: Mechanisms to Improve Data Acc...DanBrown980551
This LF Energy webinar took place June 20, 2024. It featured:
-Alex Thornton, LF Energy
-Hallie Cramer, Google
-Daniel Roesler, UtilityAPI
-Henry Richardson, WattTime
In response to the urgency and scale required to effectively address climate change, open source solutions offer significant potential for driving innovation and progress. Currently, there is a growing demand for standardization and interoperability in energy data and modeling. Open source standards and specifications within the energy sector can also alleviate challenges associated with data fragmentation, transparency, and accessibility. At the same time, it is crucial to consider privacy and security concerns throughout the development of open source platforms.
This webinar will delve into the motivations behind establishing LF Energy’s Carbon Data Specification Consortium. It will provide an overview of the draft specifications and the ongoing progress made by the respective working groups.
Three primary specifications will be discussed:
-Discovery and client registration, emphasizing transparent processes and secure and private access
-Customer data, centering around customer tariffs, bills, energy usage, and full consumption disclosure
-Power systems data, focusing on grid data, inclusive of transmission and distribution networks, generation, intergrid power flows, and market settlement data
"What does it really mean for your system to be available, or how to define w...Fwdays
We will talk about system monitoring from a few different angles. We will start by covering the basics, then discuss SLOs, how to define them, and why understanding the business well is crucial for success in this exercise.
Must Know Postgres Extension for DBA and Developer during MigrationMydbops
Mydbops Opensource Database Meetup 16
Topic: Must-Know PostgreSQL Extensions for Developers and DBAs During Migration
Speaker: Deepak Mahto, Founder of DataCloudGaze Consulting
Date & Time: 8th June | 10 AM - 1 PM IST
Venue: Bangalore International Centre, Bangalore
Abstract: Discover how PostgreSQL extensions can be your secret weapon! This talk explores how key extensions enhance database capabilities and streamline the migration process for users moving from other relational databases like Oracle.
Key Takeaways:
* Learn about crucial extensions like oracle_fdw, pgtt, and pg_audit that ease migration complexities.
* Gain valuable strategies for implementing these extensions in PostgreSQL to achieve license freedom.
* Discover how these key extensions can empower both developers and DBAs during the migration process.
* Don't miss this chance to gain practical knowledge from an industry expert and stay updated on the latest open-source database trends.
Mydbops Managed Services specializes in taking the pain out of database management while optimizing performance. Since 2015, we have been providing top-notch support and assistance for the top three open-source databases: MySQL, MongoDB, and PostgreSQL.
Our team offers a wide range of services, including assistance, support, consulting, 24/7 operations, and expertise in all relevant technologies. We help organizations improve their database's performance, scalability, efficiency, and availability.
Contact us: info@mydbops.com
Visit: https://www.mydbops.com/
Follow us on LinkedIn: https://in.linkedin.com/company/mydbops
For more details and updates, please follow up the below links.
Meetup Page : https://www.meetup.com/mydbops-databa...
Twitter: https://twitter.com/mydbopsofficial
Blogs: https://www.mydbops.com/blog/
Facebook(Meta): https://www.facebook.com/mydbops/
Essentials of Automations: Exploring Attributes & Automation ParametersSafe Software
Building automations in FME Flow can save time, money, and help businesses scale by eliminating data silos and providing data to stakeholders in real-time. One essential component to orchestrating complex automations is the use of attributes & automation parameters (both formerly known as “keys”). In fact, it’s unlikely you’ll ever build an Automation without using these components, but what exactly are they?
Attributes & automation parameters enable the automation author to pass data values from one automation component to the next. During this webinar, our FME Flow Specialists will cover leveraging the three types of these output attributes & parameters in FME Flow: Event, Custom, and Automation. As a bonus, they’ll also be making use of the Split-Merge Block functionality.
You’ll leave this webinar with a better understanding of how to maximize the potential of automations by making use of attributes & automation parameters, with the ultimate goal of setting your enterprise integration workflows up on autopilot.
inQuba Webinar Mastering Customer Journey Management with Dr Graham HillLizaNolte
HERE IS YOUR WEBINAR CONTENT! 'Mastering Customer Journey Management with Dr. Graham Hill'. We hope you find the webinar recording both insightful and enjoyable.
In this webinar, we explored essential aspects of Customer Journey Management and personalization. Here’s a summary of the key insights and topics discussed:
Key Takeaways:
Understanding the Customer Journey: Dr. Hill emphasized the importance of mapping and understanding the complete customer journey to identify touchpoints and opportunities for improvement.
Personalization Strategies: We discussed how to leverage data and insights to create personalized experiences that resonate with customers.
Technology Integration: Insights were shared on how inQuba’s advanced technology can streamline customer interactions and drive operational efficiency.
QR Secure: A Hybrid Approach Using Machine Learning and Security Validation F...AlexanderRichford
QR Secure: A Hybrid Approach Using Machine Learning and Security Validation Functions to Prevent Interaction with Malicious QR Codes.
Aim of the Study: The goal of this research was to develop a robust hybrid approach for identifying malicious and insecure URLs derived from QR codes, ensuring safe interactions.
This is achieved through:
Machine Learning Model: Predicts the likelihood of a URL being malicious.
Security Validation Functions: Ensures the derived URL has a valid certificate and proper URL format.
This innovative blend of technology aims to enhance cybersecurity measures and protect users from potential threats hidden within QR codes 🖥 🔒
This study was my first introduction to using ML which has shown me the immense potential of ML in creating more secure digital environments!
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor IvaniukFwdays
At this talk we will discuss DDoS protection tools and best practices, discuss network architectures and what AWS has to offer. Also, we will look into one of the largest DDoS attacks on Ukrainian infrastructure that happened in February 2022. We'll see, what techniques helped to keep the web resources available for Ukrainians and how AWS improved DDoS protection for all customers based on Ukraine experience
Discover the Unseen: Tailored Recommendation of Unwatched ContentScyllaDB
The session shares how JioCinema approaches ""watch discounting."" This capability ensures that if a user watched a certain amount of a show/movie, the platform no longer recommends that particular content to the user. Flawless operation of this feature promotes the discover of new content, improving the overall user experience.
JioCinema is an Indian over-the-top media streaming service owned by Viacom18.
AppSec PNW: Android and iOS Application Security with MobSFAjin Abraham
Mobile Security Framework - MobSF is a free and open source automated mobile application security testing environment designed to help security engineers, researchers, developers, and penetration testers to identify security vulnerabilities, malicious behaviours and privacy concerns in mobile applications using static and dynamic analysis. It supports all the popular mobile application binaries and source code formats built for Android and iOS devices. In addition to automated security assessment, it also offers an interactive testing environment to build and execute scenario based test/fuzz cases against the application.
This talk covers:
Using MobSF for static analysis of mobile applications.
Interactive dynamic security assessment of Android and iOS applications.
Solving Mobile app CTF challenges.
Reverse engineering and runtime analysis of Mobile malware.
How to shift left and integrate MobSF/mobsfscan SAST and DAST in your build pipeline.
The Department of Veteran Affairs (VA) invited Taylor Paschal, Knowledge & Information Management Consultant at Enterprise Knowledge, to speak at a Knowledge Management Lunch and Learn hosted on June 12, 2024. All Office of Administration staff were invited to attend and received professional development credit for participating in the voluntary event.
The objectives of the Lunch and Learn presentation were to:
- Review what KM ‘is’ and ‘isn’t’
- Understand the value of KM and the benefits of engaging
- Define and reflect on your “what’s in it for me?”
- Share actionable ways you can participate in Knowledge - - Capture & Transfer
This talk will cover ScyllaDB Architecture from the cluster-level view and zoom in on data distribution and internal node architecture. In the process, we will learn the secret sauce used to get ScyllaDB's high availability and superior performance. We will also touch on the upcoming changes to ScyllaDB architecture, moving to strongly consistent metadata and tablets.
3. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y
3
2
1
-4 -3 -2 -1 1 2 3 4 x
-1
-2
-3
4. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y
3
2
1
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
-2
-3
5. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
-2
-3
6. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
-3
7. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
-3
8. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3
9. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
10. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
C = -2 + i
11. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
C = -2 + i
12. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
13. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
14. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
15. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3
2
C 1 E
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
16. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3 NOTE: Conjugates
2 are reflected in the
C 1 E real (x) axis
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
17. The Argand Diagram
Complex numbers can be represented geometrically on an Argand
Diagram. y (imaginary axis)
3 NOTE: Conjugates
2 are reflected in the
C 1 E real (x) axis
A
-4 -3 -2 -1 1 2 3 4 x (real axis)
-1 D
A=2 -2
B = -3i -3 B
C = -2 + i
D=4-i
E=4+i
Every complex number can be represented by a unique
point on the Argand Diagram.
19. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
O x
20. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
O x
21. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
O x
22. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
y
O x x
23. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
y r x2 y2
O x x
24. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
O x x
25. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
z x2 y2
O x x
26. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
27. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
arg z z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
28. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
arg z z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
1 y
arg z tan
x
29. Mod-Arg Form
Modulus
The modulus of a complex number is the
y length of the vector OZ
z = x + iy
r 2 x2 y2
r z
y r x2 y2
arg z z x2 y2
O x x
Argument
The argument of a complex number is
the angle the vector OZ makes with the
positive real (x) axis
y
1
arg z tan arg z
x
34. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
35. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
36. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
37. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
38. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
39. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
r cos ir sin
40. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
r cos ir sin
r cos i sin
41. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
r cos ir sin
r cos i sin
The mod-arg form of z is;
42. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
r cos ir sin
r cos i sin
The mod-arg form of z is;
z r cos i sin
43. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
r cos ir sin
r cos i sin
The mod-arg form of z is;
z r cos i sin
z rcis
44. e.g . Find the modulus and argument of 4 4i
arg4 4i tan 4
4 4i 4 4
1
2 2
4
32 tan 1 1
4 2
4
Every complex number can be written in terms of its modulus and
argument
z x iy
r cos ir sin
r cos i sin
The mod-arg form of z is;
z r cos i sin
z rcis
where; r z
arg z